In this study, an effective technique for estimating parameters of autoregressive models with non-Gaussian errors has been presented. For non-Gaussian error distributions, classical techniques like maximum likelihood estimation (MLE) are found to be computationally intractable. To address this issue, an estimating function (EF) technique has been utilized, which effectively estimates the model parameters by taking advantage of the error structure. Specifically, AR(1) with logistic errors has been used, and the optimal estimating functions have been derived by constructing martingale-based estimating equations tailored to logistic errors. A hybrid AR(1)-ANN model has also been developed to integrate the strengths of both linear AR(1) and nonlinear ANN models. The robustness and efficiency of the proposed approach in parameter estimation have been investigated using simulations, comparing its mean squared error (MSE) and bias to those of MLE. The applicability of this approach has been further demonstrated using both simulated and real datasets. The results show that, in the presence of non-Gaussian errors, the EF method provides a computationally efficient and reliable alternative to classical estimation methodologies.